Detection of cosmic laments using the Candy modelmath.univ-lille1.fr/~stoica/PDF/rruji682004.pdf ·...

Detection of cosmic filaments using the Candy model

Radu S. Stoica∗

Departament de Matemàtiques, Universitat Jaume I,

Campus Riu Sec, E-12071 Castelló, Spain

Vicent J. Mart́ınez†

Observatori Astronòmic de la Universitat de València,

Apartat de correus 22085, 46075 València, Spain

Jorge Mateu‡

Departament de Matematiques, Universitat Jaume I,

Campus Riu Sec, E-12071 Castelló, Spain

Enn Saar§

Tartu Observatoorium, Tõravere, 61602 Estonia

(Dated: May 19, 2004)

Abstract

We propose to apply a marked point process to automatically delineate filaments of the large-

scale structure in redshift catalogues. We illustrate the feasibility of the idea on an example of

simulated catalogues, describe the procedure, and characterize the results. We find the distribution

of the length of the filaments, and suggest how to use this approach to obtain other statistical

characteristics of filamentary networks.

PACS numbers: 98.65.Dx, 02.70.Rr, 42.30.Sy

∗Electronic address: [email protected]†Electronic address: [email protected]‡Electronic address: [email protected]§Electronic address: [email protected]

1

I. INTRODUCTION

The large-scale structure of the Universe is studied by creating galaxy maps – positions

of thousands (a few years ago) and millions (nowadays) of galaxies in space. The angular

positions of galaxies are relatively easy to measure, but their distances can be estimated only

by measuring their recession velocities. The latter task is difficult, especially for faint distant

objects, and thus really detailed maps of galaxies have started to appear only lately. An

additional caveat is that the recession velocities contain a contribution from the dynamical

velocity of a galaxy, so the apparent distances of galaxies are in error. Such maps are called

’redshift space’ maps, but the distance errors are not as serious as to change the overall

picture of the large-scale structure.

An overview of such galaxy maps is given in [1]. As an example, we present here two

maps. The first map comes from a well-known recent galaxy survey, the Las Campanas

Redshift Survey (LCRS, [23]). This survey measured the redshifts (recession velocities) of

galaxies in six slices of width of 80◦ and of thickness of 1.5◦; a typical number of galaxies in

a slice is about 4000, and the depth of the survey is about 575 h−1Mpc[31] (corresponding

to a redshift z = 0.2 for the standard cosmological model). A map of one of the slices (the

−42◦ slice) is shown in Fig. 1, in the upper panel.The other map we show is from the most recent, ongoing survey, the Sloan Digital Sky

Survey (SDSS, [29]). This survey will measure fainter galaxies than the LCRS, will reach

deeper in space and will finally cover a full adjoint π sterradians of the sky. The data that

have been released by now consist also of separate slices; we chose a contiguous slice of 2.5◦

thick and 60◦ wide (Data Release 1, the slice with the mean ’survey coordinate’ η ≈ −23◦).The depth of the SDSS main galaxy sample is about 840 h−1Mpc (the redshift z = 0.3). The

map of the selected slice, containing 10886 galaxies, is shown in Fig. 1, in the lower panel.

The dominant feature of these maps, as of all other galaxy maps of the large-scale struc-

ture of the universe, is the network of filaments of different size and contrast, along with

relatively empty voids between the filaments. The filamentary network contains different

scales, where smaller-scale filaments are also less prominent. The gradual disappearance

of structures with increasing distance is due to the fact that the apparent luminosity of a

galaxy is the fainter the more distant it is, and in more distant regions we can observe only

a few of the brightest galaxies.

2

FIG. 1: Galaxy maps for two recent surveys, the LCRS, top panel, and the SDSS, bottom panel.

The observers (we) are situated at the bottom of the figures. Both slices are thin (the thickness of

the LCRS slice is 1.5◦ and that of the SDSS slice is 2.6◦). The scale is the same in both panels;

the depth of the SDSS slice in the right panel is 900h−1 Mpc. The filamentary network of galaxies

is clearly seen; the disappearance of structure with depth (towards the top of the figure) is caused

by luminosity selection.

3

Although the filaments are prominent, there is no good method to describe such a fil-

amentary structure. The usual second moment methods in real space or in the Fourier

space (the two-point correlation function and power spectra) do not describe well filamen-

tary structures. The method that has been used most is the minimal spanning tree (MST,

see a review in [1]). The first application of the MST formalism to describe the filamentary

networks of galaxy maps was that of [2]; many later studies have used it.

The minimal spanning tree is unique for a given point set, which is good, and it connects

all the points, which is not good. When the number of galaxies is large, the MST is rather

fuzzy, and it describes mainly the local nearest-neighbour distribution (we shall show an

example of a minimal spanning tree in sec. V). The filamentary network seen by eye combines

both local and large-scale features of the point distribution. Thus, a better notion would

be that of the skeleton, proposed recently to describe continuous density fields [19]. The

skeleton is formed by lines parallel to the gradient of the field, which connect the saddle

points to local maxima of the field. Calculating the skeleton, however, involves smoothing

the point distribution, which will introduce an extra parameter, therefore this method is not

well suited for point distributions.

We propose to use an automated method to trace filaments for realizations of point

processes, that has been shown to work well for detection of road networks in remote sensing

situations [14, 26, 27]. This method is based on the Candy model, a marked point process,

where segments serve as marks. As this is the first time such a method is used for the galaxy

distribution, we describe it in detail below. We test it also on 2-D simulated galaxy maps,

justifying our data choice. The task differs considerably from road network detection, as the

noise is larger, and we have no continuous roads, but sparsely populated ridges instead.

The present approach allows us to find the length distribution for the filaments; we give

examples of this distribution for different data samples. In this paper, we choose the Candy

process parameters by trial and error following a reversible jump process. As the method is

automated, it can also be used to estimate those parameters by using maximum likelihood

methods; these will serve then as new statistics for filament networks.

4

II. MARKED POINT PROCESSES

Let (K,B, ν) be a measure space, where K is a compact subset of R2 of strictly positiveLebesgue measure 0 < ν(K) < ∞ and B the associated Borel σ−algebra of subsets ofK. For n ∈ N let Kn be the set of all unordered configurations k = {k1, k2, . . . , kn} thatconsists of n not necessarily distinct points ki ∈ K. Let us consider the configuration spaceΩ = ∪∞n=0Kn equipped with the σ−algebra F generated by the mappings {k1, k2, . . . , kn} →∑n

i=1 1{ki ∈ B} counting the number of points in Borel sets B ∈ B. A point process on Kis a measurable map from a probability space into (Ω,F).

The reference measure is given by the unit rate Poisson process that distributes the points

in K according to a Poisson process with intensity ν.

Different characteristics or marks may be attached to the points. Under these cir-

cumstances, we consider a point process on K × M as the random sequence x ={(k1, m1), . . . , (kn, mn)} where n ∈ N0, ki ∈ K and mi ∈ M for all i = 1, . . . , n. Thecharacteristics space M is equipped with its corresponding Borel σ−algebra and the prob-ability measure νM . A marked point process X with locations in K and marks in M is a

point process on K × M such that the distribution of locations only is a point process onK.

In this case, the reference measure is the unit rate Poisson process on K × M , withthe locations distributed according to a Poisson process with intensity ν and i.i.d marks

according to νM . When the marks represent parameters of an object, such a process is

sometimes called an object point process.

The reference measure exhibits no interaction between points or objects. Indeed, we can

construct a much more complicated marked point process by specifying a probability density

with respect to the reference measure :

p(x) = α exp[−U(x)], (1)

with α the normalizing constant and U(x) the interaction energy of the system. The energy

function is written as the sum

U(x) =

q∑

j=1

∑

{xi1,...,xij}∈x

ω(j)(xi1, . . . , xij)

where ω(j) : (K × M)j → R for j = 1, . . . , q are the interaction potentials. The markedpoint processes with the probability density of the form given by (1) are known in physics

5

under the name of Gibbs point processes. If there exists a positive real C > 0 such that

U(x) − U(x ∪ {(k, m)}) ≤ log C for all (k, m) ∈ K × M the process is said to be locallystable.

This relation implies the Ruelle’s stability condition [22], which ensures the integrability of

the given probability density function. Furthermore, local stability is essential in establishing

convergence proofs for the Monte Carlo dynamics simulating such a model [9].

For our problem, y, the data to be analysed, consists of points (galaxies) spread in a

finite window K. We want to extract the filamentary structure of this data. It is natural to

consider the filaments x we want to detect as a set of random segments being the realization

of a marked point process.

The probability density of such a marked point process is given by

py(x) ∝ exp[−(Uy(x) + Ur(x))] (2)

with the terms Uy(x) and Ur(x) being the data energy and the interaction energy, respec-

tively. The first term is related to the location of the filaments among galaxies, whereas

the second is related to the geometrical properties of the filaments, playing the role of a

regularization term.

The configuration of segments composing the filamentary network is estimated by the

minimum of the total energy of the sysytem

x̂ = arg minx

{Uy(x) + Ur(x)}. (3)

In the following we will present the two components of the energy function, considerations

about the simulation of such models using the MCMC dynamics will be given and a simulated

annealing algorithm will be presented. Finally, we will apply the model to describe two-

dimensional filamentary networks of galaxies.

III. A PROBABILISTIC MODEL FOR THE FILAMENTARY STRUCTURE OF

GALAXY MAPS

A. The interaction energy : Candy model

The filaments we want to extract are composed of non-overlapping connected segments.

Locally, the curvature of one filament does not vary too much. In the data we dispose we

6

τx1

x2

x3

x4x7

x6

δ

x5

FIG. 2: Connection and alignment interaction between segments.

can notice just a few short filaments, which can be represented by isolated segments.

Under these considerations a natural choice for the interaction energy becomes the Candy

model, a marked point process simulating random networks of segments. Here, a segment

is seen as a random object ζ = (k, (θ, w, l)) that is characterized by its center location

k ∈ K and its geometrical parameters (θ, w, l) ∈ [0, π] × [wmin, wmax] × [lmin, lmax] = M ,representing its orientation, width and length respectively. The Candy model exhibits three

types of interactions between segments: connectivity, alignment and rejection.

Historically speaking, the model was introduced for the first time as a prior distribution

for thin network extraction in remotely sensed images [25–27]. Properties of the model such

as local stability and Markovianity, convergence proofs of an adapted Metropolis-Hastings

dynamics for simulating the model, as well as parameter estimation, were further investigated

in [17]. Different versions of the model were analysed and compared for the special case of

road network detection [14].

A segment has a connection region formed by the union of the two circles centered at its

extremities and of a radius rc. Two segments η = (kη, (θη, wη, lη)) and ζ = (kζ, (θζ , wζ, lζ))

are connected η ∼c ζ if only one extremity of a segment is in the connection region ofthe other segment and if ‖θη − θζ‖ ≤ τ . With respect to this definition, a segment isdoubly connected if both of its extremities are connected, singly connected if only one of its

extremities is connected and free if none of its extremities is connected. The Candy model

favors doubly connected segments whereas free and singly connected segments are penalized.

In Fig. 2 we show an example of a configuration of segments. The free segments are

x2, x4, x6 and x7, this because the segment x2 does not fullfil the orientation requirements

7

for the connection and the others do not respect the connection condition. The segments x3

and x5 are singly connected, whereas the segment x1 is doubly connected.

Similarly, the attraction region of a segment η is the union of both circles centered at

each extremity with a radius ro = lη/4. Two segments η and ζ exhibit alignment interaction

η ∼o ζ if d(kη, kζ) > 12 max{lη, lζ}, if only one extremity of a segment is in the attractionregion of the other segment, and if min{‖θη − θζ‖, π − ‖θη − θζ‖} > τ , with τ a thresholdvalue. The Candy model penalizes the segments having alignment interaction.

In the configuration shown in Fig. 2 x3 6∼o x1 and x5 6∼o x1, while the segments x2 ∼ox1 and x4 ∼o x1 because these pairs of segments exhibit high differences between theirorientations.

Connectivity is a stronger interaction than alignment. Still, as we look for the filaments

fitting the data in a random way, this last interaction gives us the possibility not to elim-

inate from the current configuration the segments with low data energy, which are almost

connected.

Every segment η is provided with a rejection region given by a circle centered in kη and of

a radius rr = lη/2. Two segments η and ζ exhibit rejection interaction if d(kη, kζ) <lη+lζ

2and

if ‖‖θη−θζ‖−π/2‖ > δ, where δ is a threshold value. The Candy model forbids configurationscontaining rejecting segments, avoiding configurations containing overlapping segments.

If d(kη, kζ) ≤ 12 max{lη, lζ} and if ‖‖θη − θζ‖ − π/2‖ ≤ δ, then the segments may crossor form a ”T” junction. The configurations with crossing segments η ∼x ζ are forbidden bythe Candy model, whereas the ”T” junctions are allowed.

Clearly, in Fig. 2 the segments x1 and x6 do not reject each other since they are far

enough, while the segments x1 and x7 do not cross, forming a ”T” junction.

For any configuration of segments x = {x1, . . . , xn} with i = 1, . . . , n, we are able now towrite for the probability density of the Candy model

pr(x) ∝{

∏n(x)i=1 exp

[

li−lmaxlmax

+ wi−wmaxwmax

]}

×γ

nd(x)d γ

nf (x)f γ

ns(x)s γ

no(x)o ×

∏

i 0 and γo ∈ (0, 1) are the model parameters, nd(x), nf (x), ns(x) are thenumbers of doubly, free and singly connected segments, and no(x) is the number of pairs

of segments which are not well aligned. In order to avoid too much small segments in the

8

configuration, the model favors segments covering a big area. Clearly the interaction energy

is obtained taking Ur(x) = − log pr(x).With respect to the classical definition of the Candy model in [17], the model described

by (4) contains differences in the definition of interactions between segments. We kept

the same name for our model, as we believe that the modifications required to apply it to

cosmological data do not change the basic premises of the classical Candy model. Con-

cerning connectivity, the present modifications were introduced in order to eliminate some

“undesired” configurations, as a segment being connected with itself or a segment being con-

nected at one extremity with both extremities of another segment. Furthermore, the new

modifications allow us to build more appropriate tailored moves for the Metropolis-Hastings

dynamics simulating the model. The rejection region was extended, as the filaments we

observe may be rather wide, hence we want to avoid overlapping of segments when the data

is good enough. This is also the reason why the width penalty was introduced. Nevertheless,

it is easy to prove that under these modifications, together with the crossing interaction the

Candy model is still locally stable and (Ripley-Kelly) Markov [21].

B. The data energy

The data energy checks whether a segment belongs to the network or not [25–27]. A

segment x is considered a part of the filament network, if its geometrical shape x̃ covers as

many galaxies as possible. Still, we want to avoid the case where segments are found in a

cloud of points rather than in a filament. To do this, we consider the shadow segments xr

and and xl – the segments situated to the right and to the left of the segment x, as in Fig. 3.

The above-mentioned case is avoided if the number of galaxies covered by x̃r and x̃l is small.

Therefore, let us define the quantity vy(x) given by

vy(x) = 2n(y ∩ x̃) − n(y ∩ x̃r) − n(y ∩ x̃l),

where n(y ∩ x̃) is the number of galaxies covered by the geometrical shape of the segmentx. Now, if the following three conditions: vy(x) ≥ 3, n(y ∩ x̃) > n(y ∩ x̃r), and n(y ∩x̃) > n(y ∩ x̃l) are simultaneously fulfilled, the data energy contribution of a segment isVy({x}) = −vy(x). If only one of the three conditions is not verified then Vy({x}) = Vmax,with Vmax > 0 a positive fixed value.

9

xi+k

xi

xirxil

i+k

i+k

l

w

FIG. 3: Locating segments in a pattern of points.

The total data energy is defined as the sum of the data energy contributions of every

segment in the configuration

Uy(x) =∑

x∈x

Vy({x}) (5)

C. Simulation dynamics and optimization

The equations (4) and (5) provide us with all the ingredients needed to construct the

Gibbs point process given by (2). The estimate of the network (3) is obtained by means of

a simulated annealing algorithm.

This algorithm iteratively samples the law [py(x)]1

T while slowly decreasing the temper-

ature T . At high positive values of temperature the space state is explored. When the

temperature goes down to zero, T → 0, the configurations of minimal energy are reached.A polynomial decreasing scheme Tk+1 = cTk with c ∈ [0.99, 1.00] may be used for cooling.

To sample from a probability density of a point process several Monte Carlo methods

are available, such as the spatial birth-and-death processes, the Metropolis-Hastings and

reversible jumps dynamics, or the much more recent exact simulation techniques as coupling

from the past or clan of ancestors [8–10, 13, 16, 18, 20]. The Candy point process exhibits

rather complicate interactions, hence the use of the spatial birth-and-death process or the

10

cited exact simulation techniques are useful in practice only for a limited range of the model

parameters. Therefore, for our present model we opted for a sampling algorithm based on

the Metropolis-Hastings dynamics. Details concerning the implementation of samplers for

the Candy model based on Metropolis-Hastings or reversible jumps processes can be found

in [17, 25–27].

IV. DATA

The Candy process and its applications have been developed for 2-D maps. So the natural

way to introduce them in cosmology is to consider 2-D cases, also. It will allow us to compare

the results, and will make it easier to understand the problems arising. Our final goal is to

apply the Candy process to describe 3-D networks of filaments, as the large-scale structure

maps fill the space. The 3-D network consists of complete filaments, as do the 2-D road

geographical road maps, so the filaments in the test data should also be complete.

The observational galaxy maps showing filaments (see Fig. 1) have mainly the geometry

of a thin slice, as those shown in the figure. Although such data have been used to study

the large-scale filamentary structure, the slices do not provide proper data for that. The

thickness of these slices is much smaller than the typical size of a filament, and although the

maps give a visual impression of filaments, the filaments we see are pieces of real filaments,

obtained by planar cuts through the real 3-D structure.

Another possibility is to use thicker slices, which can be selected, e.g., from the only

large-scale contiguous data for the moment, the 2dF survey [6]. But this choice carries

its own difficulties – thick slices give us the 2-D projection of the 3-D network, smearing

essential details.

Simulations of the formation and evolution of large-scale structure can also provide us

with galaxy maps. As a demonstration that we understand the basic features of the process,

these maps show filamentary structure. How and why an initial Gaussian random density

field develops filaments under self-gravitation, is an interesting matter, well explained by [5].

The usual simulations give us 3-D worlds, but it easy to also simulate the evolution of

structure in a 2-D world. This has been done before, to obtain better numerical resolution

(see, e.g. [3]); we used 2-D simulations to get complete cosmological networks of model

galaxies.

11

1e-08

1e-06

0.0001

0.01

1

100

10000

0.001 0.01 0.1 1 10 100

k(h/Mpc)

P3

P2

∆2

FIG. 4: The spectral density used for the 2-D simulation (P2), the corresponding spectral density

for the 3-D case (P3), and the spectral energy per unit logarithmic wavenumber interval ∆2, versus

the wavenumber k.

The present-day large-scale structure is determined, first, by the expansion history of the

cosmological model, and, secondly, by the initial density and velocity fields at the start of

the simulation. We chose the standard ’concordance’ cosmological model [28] to describe

the expansion. As the initial fields are assumed to be Gaussian random fields, they are

described by their power spectra (the spectral density of the density perturbations P (k),

where k is the module of the wave-vector; see, e.g. [1]). We chose a simple expression for the

spectral density that describes reasonably well the CDM (Cold Dark Matter) model [12] and

modified it to get the same spectral energy contribution to the variance per unit logarithmic

wavenumber interval, ∆2(k), in our 2-D world, as in the real 3-D world. In a 3-D world this

quantity is defined as

∆23(k) =1

2π2P3(k)k

3,

and in a 2-D world as

∆22(k) =1

2πP2(k)k

2;

the equality of the above quantities gives

P2(k) =k

πP3(k)

(the lower indices show the dimensionality of the space). This is the spectral density we

used, with P3(k) taken from [12]. Both the spectral densities and the spectral energy used

12

are shown in Fig. 4. As usual, the wavenumber is given in units of h/Mpc, the spectral

densities are in units of Mpc3/h3 (P3(k)), Mpc2/h2 (P2(k)), and ∆

2(k) is dimensionless.

We selected the scales and spectrum amplitudes to get a picture similar to that we see in

3-D models (the size of the patch we modelled was 128h−1Mpc, and we used a 2562 grid with

the same number of cold dark matter particles). These numbers are not really important, as

this is a mock model, anyway. Then we ran a 2-D dynamical N -body simulation, developing

the initial perturbations into large-scale structures – the present-day density and velocity

fields.

TABLE I: Parameters of the data sets: a is the cosmological expansion factor, n is the number of

galaxies, α is the void density threshold and β is the biasing amplitude.

Case a n α β

A 1.0 4127 0.5 0.20

B 0.6 4249 0.5 0.18

C 0.2 8879 1.0 0.49

These density fields describe the dark matter content of the universe. Populating model

universes with galaxies is a complex problem, but for our purposes simple recipes are suf-

ficient. We used two well-known properties of the large-scale galaxy distribution. First,

galaxies avoid large low-density regions, known as voids; we modeled this by selecting a

density threshold α (all our densities are given in the units of the mean density). In regions

with density lower than this threshold no galaxies were placed. Secondly, galaxy density is

biased in respect to the dark matter density. We found that the model galaxy distribution

resembled best the observational maps for a nonlinear biasing law:

ρgal = β√

ρCDM, ρCDM ≥ α. (6)

We chose the amplitudes α and β to produce approximately the same number of galaxies as

observed in cosmological slices of similar size.

Finally, we generated a realization of a Cox point process, using the galaxy density

given by (6) as the driving probability. In order to see how well the Candy model works

in different situations, we chose three different time moments from the simulation, with a

different filamentary structure. As the earliest of them (our case C) has a very rich set of

13

FIG. 5: Dark matter density (upper panel, logarithmic scale) and galaxy distribution (lower panel)

for the data set B.

filaments, we generated about twice as many galaxies for that data set as for other sets. As

usual in cosmology, we characterize the time moments by the value of the expansion factor

a. This factor equals unity at the present epoch and the earlier the epoch, the smaller is the

expansion factor (our universe expands). The parameters for our three data sets are given

in Table I, and the dark matter density and galaxy distributions are compared for the set

B in Fig. 5.

All the data sets are shown in Fig. 6.

14

A) 20 40 60 80 100 120

20

40

60

80

100

120

B) 20 40 60 80 100 120

20

40

60

80

100

120

C) 20 40 60 80 100 120

20

40

60

80

100

120

FIG. 6: Three sets of data

15

A. Experimental results

A simulated annealing algorithm was implemented based on Metropolis-Hastings dynam-

ics. The parameter for the cooling scheme was taken as c = 0.9995 and the initial tempera-

ture was set to 10. The algorithm was run for 107 iterations, whereas the temperature was

lowered every 103 steps.

The Candy model has a large number of parameters, and these should be chosen rather

carefully in order to get a good representation of the filaments in data. The segment

parameters (segment lengths and widths) have to be chosen to let the model filaments

follow those in data. Thus, for the first two data sets, the segment parameters were

lmin = 3, lmax = 5, wmin = 1, wmax = 2; for the third data set, smaller segments were

considered: lmin = 2, lmax = 3, wmin = 0.95, wmax = 1.05 (all distances are given in h−1 Mpc).

The interaction regions were defined by choosing the radius of the connecting region rc = 0.5

and the rejection parameter that forbids segments to cross, δ = 0.1 radians. The orientation

parameter, which limits the local curvature of filaments, was chosen as τ = 0.5 radians for

the first two data sets and as τ = 0.75 radians for the data set C.

We experimented with a large number of interaction parameters. Here we show the results

for the three sets, which give almost equally good results. The interaction parameters for

these sets are given in Table II. The optimization method was run for each data set. High

potentials were given to undesired configurations as single and free segments, badly aligned

pairs of segments with respect to the parameter τ , and badly placed segments with respect

to the data term.

TABLE II: The interaction parameters.

Sets

Parameters 1 2 3

− log γd −10 −10 −10

− log γf 8 7 7

− log γs 2 2 1

− log γo 3 3 3

Vmax 25 25 25

16

0 20 40 60 80 100 1200

20

40

60

80

100

120

0 20 40 60 80 100 1200

20

40

60

80

100

120

FIG. 7: Results obtained for the data set A: upper panel — the “best network” extraction su-

perposed on the data, lower panel — the three networks superposed. The network for the first

parameter set is shown by black lines, for the set 2 — by grey lines, and for the set 3 — by light

grey lines.

In the Figs. 7,8 and 9 we compare for each set the data and the best filament network,

and compare all three networks. We note that we do not use the periodicity of the data

— although numerical simulations are mostly periodic, the real galaxy distribution is not.

Thus it has no sense to complicate the numerical procedure.

The best set of parameters for the data set A was set 1. Examining Fig. 7 we see that

the procedure works well. All obvious filaments, which one would draw by eye, are found,

and the placement of “secondary” filaments in more sparsely populated regions is also good.

Note also that galaxy concentrations (“clusters”) do not destroy the filamentary pattern;

17

filaments usually branch in these regions.

The difference between the sets is slight, all parameter sets represent the network fairly

well. All strong filaments coincide, the difference is in the small and weak filaments, built

on a few points only. This is well seen in the lower panel, where all three networks are

superposed. The parameter set 2, e.g., generates spurious filaments in a sparsely populated

upper top of the data region, and the set 3 produces several very short isolated filaments.

On the other hand, it also provides a perfect branching point at x = 90, y = 30, that sets 1

and 2 do not find.

Figure 8 illustrates the filament networks found for the data set B. This data set has a

richer and more uniform selection of filaments than the set A. As these sets have approxi-

mately the same number of galaxies, individual filaments in the set B are more sparse and

harder to identify. Nevertheless, the method works well, especially for the parameter set 1 –

the best set; this network is shown in the upper panel of Fig. 8. There are only a few ques-

tionable short filaments, e.g., around x = 70, y = 120 and x = 10, y = 50. The parameter

set 2 generates considerably more short isolated filaments, which do not represent the data

well, and the filaments for the parameter set 3 tend to deviate in wrong directions.

The data set C has the richest set of filaments. Those are shorter and not as pronounced as

the filaments in the two first data sets — this is the way the large scale structure develops in

the universe. The early structure that the set C describes evolves by concentrating into larger

and larger clusters and filaments; small-scale structure gets weaker and disappears gradually.

In order to apply the Candy model, we had to generate about twice more galaxies for this

set than for the other two. As shown in Fig. 9, our procedure delineates the filamentary

network satisfactorily here, too, although, probably, the segments should have been smaller

yet. As seen in the upper panel for the best parameter set (set 1 in this case, too), segments

sometime jump from an obvious filament to another (e.g., at x = 47, y = 20); there is also

a tendency to form short filaments for a collection of a few points, as at x = 7, y = 60 and

x = 75, y = 107.

The parameter set 2 is in this case about as good as the set 1; it tends to miss a few

obvious filaments, however (e.g., at x = 124, y = 50), and has difficulties in resolving

interaction regions (knots in the network), see the region at x = 70, y = 110. This region

has been equally difficult to model for all three parameter sets. And, finally, the parameter

set 3 gives the worst filament placement between the three.

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0 20 40 60 80 100 1200

20

40

60

80

100

120

0 20 40 60 80 100 1200

20

40

60

80

100

120

FIG. 8: Results obtained for the data set B: upper panel — the “best network” extraction su-



grey lines.

B. Length distribution

As the Candy model is able to reconstruct the filamentary network, given a point process

(galaxy map), the collection of its parameters can be considered as a description of the

network. When determined from the data by a likelihood procedure, they can serve as

statistics of the network. But there are simple statistics we can already study; the simplest

one is the probability distribution of the lengths of individual filaments (sets of connected

segments). A similar problem, that of the length of the largest filament, has been addressed

recently, using a pixel-based method to define filaments and finding the pixel size, where the

19

0 20 40 60 80 100 1200

20

40

60

80

100

120

0 20 40 60 80 100 1200

20

40

60

80

100

120

FIG. 9: Results obtained for the data set C: upper panel — the “best network” extraction su-



grey lines.

filaments are yet statistically significant [4]. As filaments delineate voids, the distribution

of the length of filaments is also connected with the distribution of void sizes. This subject

has a long history; see, e.g. [1] and an interesting recent theoretical paper [24].

Comparison of the length histograms in Fig. 10 reveals a series of peaks, several distinct

characteristic lengths in the filamentary network[32]. These peaks are especially promi-

nent for the data sets A and C, and smeared out for the set B. Also, the locations of the

peaks do not depend much on the specific parameter set, the peaks more or less coincide.

And although the sample sizes are a bit too small for the first two data sets to draw firm

20

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 10 20 30 40 50 60 70 80 90

f(l)

l

A

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 10 20 30 40 50 60 70 80 90

f(l)

l

B

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 10 20 30 40 50 60 70 80 90

f(l)

l

C

FIG. 10: Filament length distribution histograms for the three data sets (marked in the panels).

Solid lines indicates the parameter set 1, dashed lines – set 2, and dotted lines – set 3.

conclusions, inspection of the integral probability distributions shows that the peaks are

real.

Another feature of the distributions is their long wings – the lengths of filaments reach

about 90, almost the full size of the box. Inspection of Figs. 7,8 and 9 shows that long

filaments are those that pass through the branching points and are really collections of several

filaments. So, we have to find in future a recipe that would locate the branching points and

break the filaments; otherwise we shall loose connection with the void distribution. For the

histograms in Fig. 10 this means that there would be additional contributions to the 20-40

length range, which are presently missing.

21

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 10 20 30 40 50 60 70 80 90

f(l)

l

FIG. 11: Combined filament length distribution histograms for the three data sets. Solid line

indicates the data set A, dashed line – set B, and dotted line – set C.

As the locations of the peaks almost do not depend on the parameter set used, we

combined the length data for the three parameter sets together. These distributions for the

three data sets are compared in Fig. 11. Thus these peaks are significant, revealing discrete

scales in the data. We also see that the overall length distribution is shifted to the shorter

sizes for the set C, compared with the set A.

V. OTHER APPROACHES

There exist only a few methods to describe the observed filamentary networks of galaxies.

The best known approach is that of minimal spanning trees (MST) [2]. The minimal span-

ning tree connects all data points, and its length distribution function describes mainly the

nearest-neighbour distance distributions, not the large-scale network we see. The MST has

to connect also all points in clusters, while the Candy model can be tuned to ignore them (as

we have seen, clusters become usually branching points of the filament network in the Candy

model). The differences between the MST and the Candy model can be well seen in Fig. 12.

Nevertheless, the MST has been extensively used to describe the filamentary network; as

the Candy model looks much better, it would probably lead to a better description of the

cosmic filaments.

Using a technique based on multiscale geometric analysis Arias-Castro et al. [30] have

shown how filaments can be detected, when they are embedded in a uniformly distributed

background of points. This algorithm is specially focused for finding hidden filamentary

22

0 20 40 60 80 100 1200

20

40

60

80

100

120

0

20

40

60

80

100

120

0 20 40 60 80 100 120

FIG. 12: A Candy model (upper panel) and the minimal spanning tree (lower panel) for the same

set of data (set A)).

patterns in images or nearly Poissonian point processes. Our approach is different and is

better suited for finding many filaments in clustered point processes.

Another, more recent approach to describe filaments [4] proceeds by binning the map

(calculating a density field), and using Minkowski functionals of the isodensity contours to

estimate filamentarity of the objects. While this approach will classify all objects, it has

two free parameters, the smoothing length (size of the density bin), and the isodensity level.

True, in some respect our approach is similar to that, as the segments of the Candy model

have a finite width (we are also estimating a density field). But our density estimator is

anisotropic and adaptive, in principle, and we trace only filamentary structures.

A third approach that is also based on a density field, is to determine the saddle points

23

and to build a network of field lines (directed along the gradient of the field), connecting

saddle points with local maxima – the skeleton [19]. This approach could reconstruct well

the cellular network of filaments (so far it has been applied to study the pixelized cosmic

microwave background data [7]), but it will also depend on the density estimation procedure.

And, as the authors say, this approach is computationally complex.

VI. CONCLUSION AND PERSPECTIVES

The parameter values for our method were chosen after several trials and errors. Under

these circumstances, parameter estimation using Monte Carlo maximum likelihood methods

may be considered [9, 17]. These parameters could then be considered as statistics describing

the filamentary network. These will certainly be much better suited for this task than the

moments of the density distribution in real or in Fourier space, which are commonly used

in cosmology.

The data term is a very simple test. Much more sophisticated techniques as testing

the alignment of the points covered by a segment, or statistical tests such as the complete

randomness test need investigation.

To our knowledge there is no proof for the existence of an optimal cooling scheme when

using Metropolis-Hastings dynamics for simulating point processes in a simulated annealing

algorithm. There is such a proof for the spatial birth-and-death process, but in practice the

authors sample the model using a fixed cold temperature [15]. The choice we opted for, a

slow polynomial decreasing scheme, does not guarantee that the global optimum is reached.

But overall, as we have seen, the results are good, the Candy model can be tuned to

trace well the filamentary network. And it can be naturally generalized to describe the real

3-D filamentary networks of galaxy maps; see [11]. As we already said above, it can also

be considered as a tool to provide statistics of filamentary networks. These are the future

directions of our work.

VII. ACKNOWLEDGEMENTS

Enn Saar and Radu Stoica want to thank the hospitality of the Observatori Astronòmic de

la Universitat de València where part of this work was done. This work has been supported

24

by Valencia University through a visiting professorship for Enn Saar, by the Spanish MCyT

project AYA2003-08739-C02-01 (including FEDER), by the Generalitat Valenciana project

GRUPOS03/170, and by the Estonian Science Foundation grant 4695. We thank Rien van

de Weygaert for his code to calculate the MST. The work of Jorge Mateu and Radu Stoica

was carried out, respectively, under the grants BFM2001-3286 of the Spanish MCyT and

SB2001-0130 from MECD.

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